1) The document discusses polynomials and their properties. It defines what a polynomial is - an expression with terms containing variables and coefficients.
2) It introduces the concept of the degree of a polynomial and defines it as the highest power of the variable terms. Constant polynomials have a degree of zero.
3) The Remainder Theorem is discussed, stating that when dividing a polynomial p(x) by a linear polynomial x-a, the remainder is equal to the value of p(a).
2. 1. Introduction
You have studied algebraic expressions, their addition,
subtraction, multiplication and division in earlier classes.
You also have to factories some algebraic expressions. You
may recall the algebraic identities:
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
and (x 2- y2 = (x + y) (x - y)
and their use in factorization. In this chapter, we shall start
our study with a particular type of algebraic expression,
called polynomial, and terminology relation to it. We shall
also study the Remainder Theorem and Factor Theorem
and their use in the factorization polynomials. In addition
to the above, we shall study some more algebraic identities
and their use in factorization and in evaluating some given
expressions.
3. 2 Polynomials in one Variable
Let us begin by recalling that a variable is denoted by a symbol that
can take any real value. We use the letter x, y, z, etc. to denote
variables. Notice that 2x, 3x, - x, - 1/2x are algebraic expressions. All
these expressions are of the form (a constant) x (a variable) and we
do not know what the constant is. In such cases, we write the
constant as a, b, c, etc. So the expression will be ax say.
In the polynomial x2 + 2x, the expression x2 and 2x are called the
terms.
Each term of a polynomial has coefficient.
The constant polynomial 0 is called the zero polynomial.
The degree of a non-zero constant polynomial is zero.
4. 3 Zeroes of a Polynomial
Consider the polynomial p(x) = 5x -2x2 + 3x - 2.
If we replace x by I everywhere in p(x), we get
p(1) = 5 x (1)3 – 2 x (1)2 + (1) – 2
= 5 – 2 + 3 – 2
= 4
So, we say that the value of p(x) at x = 1 is 4.
Similarly, p(0) =5(0)3 – 2(0)2 = 3(0) -2
Can you find p(-1)?
5. 4 Reminder Theorem
Let us consider two numbers 15 and 6. You know that when we
divide 15 by 6, we get the quotient 2 remainder 3. Do you
remember how this fact is expressed? We write 15 as.
15 = (2 x 6) + 3
We observe that the remainder 3 is less than the divisor 6.
Similarly, if we divide 12 by 6,we get
12 = (2 x 6) + 0
What is the remainder here? Here the remainder is 0, and we say
that 6 is a factor of 12 or 12 is a multiple of 6.
Now, the question is can we divide one polynomial by another? T
start with, let us try and do this when the divisor in a monomial. So,
let us divide the polynomial 2x3 + x2 + x by the monomial x.
We have (2x3 + x2 + x) ÷ x = 2x3 + x2 + x
x x x
= 2x2 + x +1
6. Remainder Theorem
Let p(x) be any polynomial of degree greater than or equal to one and let
a be any real number. If p(x) is divided by the linear polynomial x – a, then
the remainder is p(a).
Proof:
Let p(x) be any polynomial with degree greater than or equal to 1.
Suppose that when p(x) is divided by x – a, the quotient is q(x) and the
remainder is r(x), i.e.,
p(x) = (x – a) q(x) + r(x)
Since the degree of x – a is 1 and the degree of r(x) is less than the
degree of x – a, the degree of r(x) =0. This means that r(x) is of a constant,
say r.
So, for every value of x, r(x) = r.
Therefore, p(x) = (x - a) q(x) ÷ r
In particular, if x = a, this equation gives us
p(a) = (a - a) q(a) ÷ r
= r.
7. Factorisation of Polynomials
Let us now look at the situation of Example 10 above more
closely. It tells us that since the remainder,
q 1 =0, (2t + 1) is a factor of q(t), i.e., q(t) = (2t + 1) g(t)
2
for some polynomial g(t). This is a particular case of the
following theorem.
Factor Theorem:
If p(x) is a polynomial of degree n > 1 and a is any real
number then
•x –a is a factor of p(a) = 0, and
•p(a) = 0, if x – a is a factor of p(x).
8. Algebraic Identities
From your earlier classes, you may recall that an algebraic
identity is an algebraic equation that is true for all values of
the variables occurring in it. You have studied the following
algebraic identities in earlier classes:
Identity I : (x + y)2 = x2 + 2xy + y2
Identity II : (x - y)2 = x2 - 2xy + y2
Identity III : x2 + y2 = (x + y) (x - y)
Identity IV : (x + a) (x + b) = x2 + (a + b)x + ab
9. Summary
•A polynomial of one term is called a monomial.
•A polynomial of two term is called a binomial.
•A polynomial of three term is called a trinomial.
Thank You…..
